re ections on mathematics · 2012. 12. 13. · chapter 1 isometries and since you know you cannot...

Reflections on MathematicsMath 1118: Spring 2013

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Preface

These notes are designed with architecture and design students in mind. It isour hope that the reader will find these notes both interesting and challenging.In particular, there is an emphasis on the ability to communicate mathematicalideas. Three goals of these notes are:

• To enrich the reader’s understanding of geometry and algebra. We hopeto show the reader that geometry and algebra are deeply connected.

• To place an emphasis on problem solving. The reader will be exposed toproblems that “fight-back.” Worthy minds such as yours deserve worthyopponents. Too often mathematics problems fall after a single “trick.”Some worthwhile problems take time to solve and cannot be done in asingle sitting.

• To challenge the common view that mathematics is a body of knowledgeto be memorized and repeated. The art and science of doing mathematicsis a process of reasoning and personal discovery followed by justificationand explanation. We wish to convey this to the reader, and sincerely hopethat the reader will pass this on to others as well.

In summary—you, the reader, must become a doer of mathematics. To this end,many questions are asked in the text that follows. Sometimes these questionsare answered, other times the questions are left for the reader to ponder. To letthe reader know which questions are left for cogitation, a large question markis displayed:

?The instructor of the course will address some of these questions. If a questionis not discussed to the reader’s satisfaction, then I encourage the reader to puton a thinking-cap and think, think, think! If the question is still unresolved, goto the World Wide Web and search, search, search!

This document is open-source. It is licensed under the Creative CommonsAttribution-NonCommercial-ShareAlike (CC BY-NC-SA) License. Loosely speak-ing, this means that this document is available for free. Anyone can get a freecopy of this document (source or PDF) from the following site:

http://www.math.osu.edu/~snapp/1118/

Please report corrections, suggestions, gripes, complaints, and criticisms to BartSnapp at: [email protected]

Thanks and Acknowledgments

A brief history of this document: In 2009, Greg Williams, a Master of Arts inTeaching student at Coastal Carolina University, worked with Bart Snapp toproduce an early draft of the chapter on isometries. In the summer and fall of2011, Bart Snapp wrote the remaining chapters of this set of notes.

Much thanks goes to Herb Clemens, Vic Ferdinand, Betsy McNeal, andDaniel Shapiro who introduced me to ideas which have been used in this course.

Contents

1 Isometries 11.1 Matrices as Functions . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 The Algebra of Matrices . . . . . . . . . . . . . . . . . . . . . . . 141.2.1 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . 141.2.2 Compositions of Matrices . . . . . . . . . . . . . . . . . . 151.2.3 Mixing and Matching . . . . . . . . . . . . . . . . . . . . 17

1.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 221.3.2 Groups of Rotations . . . . . . . . . . . . . . . . . . . . . 231.3.3 Groups of Reflections . . . . . . . . . . . . . . . . . . . . 241.3.4 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 25

2 Folding and Tracing Constructions 292.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Anatomy of Figures . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.1 Lines Related to Triangles . . . . . . . . . . . . . . . . . . 372.2.2 Circles Related to Triangles . . . . . . . . . . . . . . . . . 37

2.3 Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Another Dimension 493.1 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1.1 Tessellations and Art . . . . . . . . . . . . . . . . . . . . . 503.2 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Platonic Solids . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 String Art 684.1 I’m Seeing Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Drawing Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.1 Envelope of Tangents . . . . . . . . . . . . . . . . . . . . 734.2.2 Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . . . 74

A Activities 81

References and Further Reading 129

Index 130

Chapter 1

Isometries

And since you know you cannot see yourself, so well as by reflection,I, your glass, will modestly discover to yourself, that of yourselfwhich you yet know not of.

—William Shakespeare

1.1 Matrices as Functions

We’re going to discuss some basic functions in geometry. Specifically, we willtalk about translations, reflections, and rotations. To start us off, we need alittle background on matrices.

Question What is a matrix?

You might think of a matrix as just a jumble of large brackets and numbers.However, we are going to think of matrices as functions. Just as we write f(x)for a function f acting on a number x, we’ll write:

Mp = q

to represent a matrix M mapping point p to point q. A point p is often repre-sented as an ordered pair of coordinates, p = (x, y). However, to make thingswork out nicely, we need to write our points all straight and narrow, with alittle buddy at the end:

(x, y)⇝

xy1

Throughout this chapter, we will abuse notation slightly, freely interchangingseveral notations for a point:

p↭ (x, y)↭

xy1

1

1.1. MATRICES AS FUNCTIONS

With this in mind, our work will be done via matrices and points that look likethis:

M =

a b cd e f0 0 1

and p =xy1

Now recall the nitty gritty details of matrix multiplication:

Mp =

a b cd e f0 0 1

xy1

=

ax+ by + c · 1dx+ ey + f · 10 · x+ 0 · y + 1 · 1

=

ax+ by + cdx+ ey + f1

Question Fine, but what does this have to do with geometry?

In this chapter we are going to study a special type of functions, calledisometries. These are function that preserves distances. Let’s see what wemean by this:

Definition An isometry is a function M that maps points in the plane toother points in the plane such that

d(p,q) = d(Mp,Mq),

where d is the distance function.

Question How do you compute the distance between two points again?

?

We’re going to see that several ideas in geometry, specifically translations, re-flections, and rotations which all seem very different, are actually all isometries.Hence, we will be thinking of these concepts as matrices.

1.1.1 Translations

Of all the isometries, translations are probably the easiest. With a translation,all we do is move our object in a straight line, that is, every point in the planeis moved the same distance and the same direction. Let’s see what happens to

2

CHAPTER 1. ISOMETRIES

Louie Llama when he is translated:

Pretty simple eh? We can give a more “mathematical” definition of a trans-lation involving our newly-found knowledge of matrices! Check it:

Definition A translation, denoted by T(u,v), is a function that moves everypoint a given distance u in the x-direction and a given distance v in the y-direction. We will use the following matrix to represent translations:

T(u,v) =

1 0 u0 1 v0 0 1

Example Consider the point p = (−3, 2). Use a matrix to translate p 5 unitsright and 4 units down.

3


Here is how you do it:

T(5,−4)p =

1 0 50 1 −40 0 1

−321

=

−3 + 0 + 50 + 2− 40 + 0 + 1

=

2−21

Hence, we end up with the point (2,−2). But you knew that already, didn’tyou?

Question Can you demonstrate with algebra why translations are isometries?

?

Question We know how to translate individual points. How do we moveentire figures and other funky shapes?

?

1.1.2 Reflections

The act of reflection has fascinated humanity for millennia. It has a strong effecton our perception of beauty and has a defined place in art—not to mention howuseful it is for the application of make-up. Here is our definition of a reflection:

Definition The reflection across a line ℓ, denoted by Fℓ, is the function thatmaps a point p to a point Fℓp such that:

(1) If p is on ℓ, then Fℓp = p.

(2) If p is not on ℓ, then ℓ is the perpendicular bisector of the segment con-necting p and Fℓp.

You might be saying, “Huh?” It’s not as hard as it looks. Check out this

4


picture of the situation, again Louie Llama will help us out:

A Collection of Reflections

We are going to begin with a trio of reflections. We’ll start with a horizontalreflection across the y-axis. Using our matrix notation, we write:

Fx=0 =

−1 0 00 1 00 0 1

The next reflection in our collection is a vertical reflection across thex-axis. Using our matrix notation, we write:

Fy=0 =

1 0 00 −1 00 0 1

The final reflection to add to our collection is a diagonal reflection across theline y = x. Using our matrix notation, we write:

Fy=x =

0 1 01 0 00 0 1

Example Consider the point p = (3,−1). Use a matrix to reflect p across the

5


line y = x.


Fy=xp =

0 1 01 0 00 0 1

3−11

=

0− 1 + 03 + 0 + 00 + 0 + 1

=

−131

Hence we end up with the point (−1, 3).

Question Let p be some point in Quadrant I of the (x, y)-plane. Whatreflection will map this point to Quadrant II? What about Quadrant IV? Whatabout Quadrant III?

?Question Can you demonstrate with algebra why each of our reflectionsabove are isometries?

?

1.1.3 Rotations

Imagine that you are on a swing set, going higher and higher until you areactually able to make a full circle1. At the point where you are directly above

1Face it, I think we all dreamed of doing that when we were little—or in my case, lastweek.

6


where you would be if the swing were at rest, where is your head, comparatively?Your feet? Your hands?

Rotations should bring circles to mind. This is not a coincidence. Check outour definition of a rotation:

Definition A rotation of θ degrees about the origin, denoted by Rθ, is afunction that maps a point p to a point Rθp such that:

(1) The points p and Rθp are equidistant from the origin.

(2) An angle of θ degrees is formed by p, the origin, and Rθp.

Louie Llama, can you do the honors?

WARNING Positive angles denote a counterclockwise rotation. Negative an-gles denote a clockwise rotation.

Looking back on trigonometry, there were some angles that kept on comingup. Some of these were 90◦, 60◦, and 45◦. We’ll focus on these angles too.

R90 =

0 −1 01 0 00 0 1

R60 = 12 −

√3

2 0√32

12 0

0 0 1

R45 = 1√2 −1√2 01√

21√2

0

0 0 1

Example Consider the point p = (4,−2). Use a matrix to rotate p 60◦ about

7


the origin.


R60p =

12 −√3

2 0√32

12 0

0 0 1

4−2

1

=

2 +√3 + 02√3− 1 + 00 + 0 + 1

=

2 +√32√3− 11

Hence, we end up with the point (2 +

√3, 2

√3− 1).

Question Do the numbers in the matrices above look familiar? If so, why?

?Question How do you rotate a point 180 degrees?

?Question Can you demonstrate with algebra why our rotations above areisometries?

?

8


Problems for Section 1.1

(1) How do you compute the distance between two points p and q in theplane?

(2) Use algebra to explain why:

d(p,q) = d(p− q,o) = d(o,p− q)

where o = (0, 0).

(3) What is an isometry?

(4) What is a translation?

(5) What is a rotation?

(6) What is a reflection?

(7) Reflecting back on this chapter, suppose I translate a point p to p′. Doesit make any difference if I move the point p along a wiggly path

or a straight path? Explain your reasoning.

(8) Reflecting back on this chapter, is a rotation the continuous act of movinga point through an angle around some fixed point, or is it just a finalpicture compared to the initial one? Explain your reasoning.

(9) In the vector illustrator Inkscape there is an option to transform an imagevia a “matrix.” If you select this tool, you are presented with 6 boxes tofill in with numbers:

Use what you’ve learned in this chapter to make a guess as to how thistool works.

(10) In what direction does a positive rotation occur?

(11) Is a 270◦ rotation the same as a −90◦ rotation? Explain your reasoning.

(12) Consider the following matrix:

M =

1 0 00 1 00 0 1

Is M an isometry? Explain your reasoning.

9



M =

0 0 82 1 00 0 1



M =

0 −1 0−1 0 00 0 1



M =

0 2 0−3 0 00 0 1


M =

1 2 11 1 20 0 1


(17) Use a matrix to translate the point (−1, 6) three units right and two unitsup. Sketch this situation and explain your reasoning.

(18) The matrix T(−2,6) was used to translate the point p to (−1,−3). Whatis p? Sketch this situation and explain your reasoning.

(19) Use a matrix to reflect the point (5, 2) across the x-axis. Sketch thissituation and explain your reasoning.

(20) Use a matrix to reflect the point (−3, 4) across the y-axis. Sketch thissituation and explain your reasoning.

(21) Use a matrix to reflect the point (−1, 1) across the line y = x. Sketch thissituation and explain your reasoning.

(22) Use a matrix to reflect the point (1, 1) across the line y = x. Sketch thissituation and explain your reasoning.

(23) The matrix Fy=0 was used to reflect the point p to (4, 3). What is p?Explain your reasoning.

10


(24) The matrix Fy=0 was used to reflect the point p to (0,−8). What is p?Explain your reasoning.

(25) The matrix Fx=0 was used to reflect the point p to (−5,−1). What is p?Explain your reasoning.

(26) The matrix Fy=x was used to reflect the point p to (9,−2). What is p?Explain your reasoning.

(27) The matrix Fy=x was used to reflect the point p to (−3,−3). What is p?Explain your reasoning.

(28) Considering the point (3, 2), use a matrix to rotate this point 60◦ aboutthe origin. Sketch this situation and explain your reasoning.

(29) Considering the point (√2,−

√2), use a matrix to rotate this point 45◦

about the origin. Sketch this situation and explain your reasoning.

(30) Considering the point (−7, 6), use a matrix to rotate this point 90◦ aboutthe origin. Sketch this situation and explain your reasoning.

(31) Considering the point (−1, 3), use a matrix to rotate this point 0◦ aboutthe origin. Sketch this situation and explain your reasoning.

(32) Considering the point (0, 0), use a matrix to rotate this point 120◦ aboutthe origin. Sketch this situation and explain your reasoning.

(33) Considering the point (1, 1), use a matrix to rotate this point −90◦ aboutthe origin. Sketch this situation and explain your reasoning.

(34) The matrix R90 was used to rotate the point p to (2,−5). What is p?Explain your reasoning.

(35) The matrix R60 was used to rotate the point p to (0, 2). What is p?Explain your reasoning.

(36) The matrix R45 was used to rotate the point p to (−12 ,52 ). What is p?

Explain your reasoning.

(37) The matrix R−90 was used to rotate the point p to (4, 3). What is p?Explain your reasoning.

(38) If someone wanted to plot the graph of y = x2, they might start by fillingin the following table:

x x2

01

−12

−23

−3

11


Reflect each point you obtain from the table above about the line y = x.Give a plot of this situation. What curve do you obtain? What is thisnew curve’s relationship to y = x2? Explain your reasoning.

(39) Some translation T was used to map point p to point q. Given p = (1, 2)and q = (3, 4), find T and explain your reasoning.

(40) Some translation T was used to map point p to point q. Given p = (−2, 3)and q = (2, 3), find T and explain your reasoning.

(41) Some reflection F was used to map point p to point q. Given p = (1, 4)and q = (1,−4), find F and explain your reasoning.

(42) Some reflection F was used to map point p to point q. Given p = (5, 0)and q = (0, 5), find F and explain your reasoning.

(43) Some rotation R was used to map point p to point q. Given p = (3, 0)and q = (0, 3), find R and explain your reasoning.

(44) Some rotation R was used to map point p to point q. Given p = (√2,√2)

and q = (0, 2), find R and explain your reasoning.

(45) Some matrix M maps

(0, 0) 7→ (0, 0),(1, 0) 7→ (3, 0),(0, 1) 7→ (0, 5).

Find M and explain your reasoning.


(0, 0) 7→ (−1, 1),(1, 0) 7→ (3, 0),(0, 1) 7→ (0, 5).



(0, 0) 7→ (1, 1),(1, 0) 7→ (2, 1),(0, 1) 7→ (1, 2).



(0, 0) 7→ (2, 2),(1, 1) 7→ (3, 3),

(−1, 1) 7→ (1, 3).


12



(0, 0) 7→ (0, 0),(1, 1) 7→ (0, 3),

(−1, 1) 7→ (5, 0).



(0, 0) 7→ (1, 2),(1, 1) 7→ (−3, 1),

(−1, 1) 7→ (2,−3).


13

1.2. THE ALGEBRA OF MATRICES

1.2 The Algebra of Matrices

1.2.1 Matrix Multiplication

We know how to multiply a matrix and a point. Multiplying two matrices is asimilar procedure:a b cd e f

g h i

j k lm n op q r

=aj + bm+ cp ak + bn+ cq al + bo+ crdj + em+ fp dk + en+ fq dl + eo+ frgj + hm+ ip gk + hn+ iq gl + ho+ ir

Variables are all good and well, but let’s do this with actual numbers. Con-

sider the following two matrices:

M =

1 2 34 5 67 8 9

and I =1 0 00 1 00 0 1

Let’s multiply them together and see what we get:

MI =

1 2 34 5 67 8 9

1 0 00 1 00 0 1

=

1 · 1 + 2 · 0 + 3 · 0 1 · 0 + 2 · 1 + 3 · 0 1 · 0 + 2 · 0 + 3 · 14 · 1 + 5 · 0 + 6 · 0 4 · 0 + 5 · 1 + 6 · 0 4 · 0 + 5 · 0 + 6 · 17 · 1 + 8 · 0 + 9 · 0 7 · 0 + 8 · 1 + 9 · 0 7 · 0 + 8 · 0 + 9 · 1

=

1 2 34 5 67 8 9

= M

Question What is IM equal to?

?

It turns out that we have a special name for I. We call it the identitymatrix.

WARNING Matrix multiplication is not generally commutative. Check itout:

F =

1 0 00 −1 00 0 1

and R =0 −1 01 0 00 0 1

14


When we multiply these matrices, we get:

FR =

1 0 00 −1 00 0 1

0 −1 01 0 00 0 1

=

1 · 0 + 0 · 1 + 0 · 0 1 · (−1) + 0 · 0 + 0 · 0 1 · 0 + 0 · 0 + 0 · 10 · 0 + (−1) · 1 + 0 · 0 0 · (−1) + (−1) · 0 + 0 · 0 0 · 0 + (−1) · 0 + 0 · 10 · 0 + 0 · 1 + 1 · 0 0 · (−1) + 0 · 0 + 1 · 0 0 · 0 + 0 · 0 + 1 · 1

=

0 −1 0−1 0 00 0 1

On the other hand, we get:

RF =

0 1 01 0 00 0 1

Question Can you draw some nice pictures showing geometrically that ma-trix multiplication is not commutative?

?Question Is it always the case that (LM)N = L(MN)?

?

1.2.2 Compositions of Matrices

It is often the case that we wish to apply several isometries successively to apoint. Consider the following:

M =

a b cd e f0 0 1

N =g h ij k l0 0 1

and p =xy1

Now let’s compute

M(Np) =

a b cd e f0 0 1

g h ij k l0 0 1

xy1

=

a b cd e f0 0 1

gx+ hy + ijx+ ky + l1

=

agx+ ahy + ai+ bjx+ bky + bl + cdgx+ dhy + di+ ejx+ eky + el + f1

Now you compute (MN)p and compare what you get to what we got above.

15


Compositions of Translations

A composition of translations occurs when two or more successive translationsare applied to the same point. Check it out:

T(5,−4)T(−3,2) =

1 0 50 1 −40 0 1

1 0 −30 1 20 0 1

=

1 0 20 1 −20 0 1

= T(5+(−3),(−4)+2)

= T(2,−2)

Theorem 1 The composition of two translations T(u,v) and T(s,t) is equal tothe translation T(u+s,v+t).

Question How do you prove the theorem above?

?Question Can you give a single translation that is equal to the followingcomposition?

T(−7,5)T(0,−6)T(2,8)T(5,−4)

?Question Are compositions of translations commutative? Are they associa-tive?

?

Compositions of Reflections

A composition of reflections occurs when two or more successive reflections areapplied to the same point. Check it out:

Fy=0Fy=x =

1 0 00 −1 00 0 1

0 1 01 0 00 0 1

=

0 1 0−1 0 00 0 1

Question Is the composition Fy=0Fy=x still a reflection?

16


?

Question Are compositions of reflections commutative? Are they associa-tive?

?

Compositions of Rotations

A composition of rotations occurs when two or more successive rotations areapplied to the same point. Check it out:

R60R60 =

12 −√3

2 0√32

12 0

0 0 1

12 −

√3

2 0√32

12 0

0 0 1

=

−12 −√3

2 0√32

−12 0

0 0 1

Theorem 2 The product of two rotations Rθ and Rφ is equal to the rotationRθ+φ.

From this we see that:

R120 =

−12 −√3

2 0√32

−12 0

0 0 1

Question What is the rotation matrix for a 360◦ rotation? What about a405◦ rotation?

?

Question Are compositions of rotations commutative? Are they associative?

?

1.2.3 Mixing and Matching

Life gets interesting when we start composing translations, reflections, and ro-tations together. First we’ll compose a reflection with a rotation:

17


Fy=0R60 =

1 0 00 −1 00 0 1

12 −

√3

2 0√32

12 0

0 0 1

=

12 −√3

2 0

−√32 −

12 0

0 0 1

Question Does this result look familiar?

?Now how about a rotation composed with a translation:

R90T(3,−4) =

0 −1 01 0 00 0 1

1 0 30 1 −40 0 1

=

0 −1 41 0 30 0 1

Question Does R90T(3,−4) = T(3,−4)R90?

?Question Find a matrix that represents the reflection Fy=−x.

I’ll take this one. Note that

Fy=−x = R180Fy=x

= R90R90Fy=x

=

0 −1 01 0 00 0 1

0 −1 01 0 00 0 1

0 1 01 0 00 0 1

=

0 −1 0−1 0 00 0 1

OK looks good, but you, the reader, are going to have to check the abovecomputation yourself.

Question How do we deal with reflections that are not across the lines y = 0,x = 0, or y = x? How would you reflect points across the line y = 1?

?

18



(1) Give a single translation that is equal to T(−3,2)T(5,−1). Explain yourreasoning.

(2) Consider the two translations T(−4,8) and T(4,−8). Do these commute?Explain your reasoning.

(3) Give a single reflection that is equal to Fx=0Fy=0. Sketch this situationand explain your reasoning.

(4) Given any point p = (x, y), express T(4,2)T(6,−5)p as T(u,v)p for somevalues of u and v. Sketch this situation and explain your reasoning.

(5) Give a matrix for R−45. Explain your reasoning.

(6) Give a matrix for R−60. Explain your reasoning.

(7) Sam suggests that:

R−90 =

0 1 0−1 0 00 0 1

Why does he suggest this? Is it even true? Explain your reasoning.

(8) Give a matrix for Fy=−x. Explain your reasoning.

(9) Given the point p = (−4, 2), use matrices to compute Fy=0Fy=xp. Sketchthis situation and explain your reasoning.

(10) Given the point p = (5, 0), use matrices to compute Fy=xFy=−xp. Sketchthis situation and explain your reasoning.

(11) Give a single rotation that is equal to R45R60. Explain your reasoning.

(12) Given the point p = (1, 3), use matrices to compute R45R90p. Sketch thissituation and explain your reasoning.

(13) Given the point p = (−7, 2), use matrices to compute R45R−45p. Sketchthis situation and explain your reasoning.

(14) Given the point p = (−2, 5), use matrices to compute R90R−90R360p.Sketch this situation and explain your reasoning.

(15) Given the point p = (5, 4), use matrices to compute Fy=0T(2,−4)p. Sketchthis situation and explain your reasoning.

(16) Given the point p = (−1, 6), use matrices to compute R45T(0,0)p. Sketchthis situation and explain your reasoning.

(17) Given the point p = (11, 13), use matrices to compute T(−6,−3)R135p.Sketch this situation and explain your reasoning.

19


(18) Given the point p = (−7,−5), use matrices to compute R540Fx=0p. Sketchthis situation and explain your reasoning.

(19) Give a composition of matrices that will take a point and reflect it acrossthe x-axis and then rotate the result 90◦ around the origin. Sketch thissituation and explain your reasoning.

(20) Give a composition of matrices that will take a point and translate it threeunits up and 2 units left and then rotate it 90◦ clockwise around the origin.Sketch this situation and explain your reasoning.

(21) Give a composition of matrices that will take a point and rotate it 270◦

around the origin, reflect it across the line y = x, and then translate theresult down 5 units and 3 units to the right. Sketch this situation andexplain your reasoning.

(22) Give a composition of translations and any of the following matrices

{Fx=0,Fy=0,Fy=x,R90,R60,R45}

that will take a point and reflect it across the line x = 1. Sketch thissituation and explain your reasoning.


{Fx=0,Fy=0,Fy=x,R90,R60,R45}

that will take a point and reflect it across the line y = −4. Sketch thissituation and explain your reasoning.


{Fx=0,Fy=0,Fy=x,R90,R60,R45}

that will take a point and reflect it across the line y = x+ 5. Sketch thissituation and explain your reasoning.


{Fx=0,Fy=0,Fy=x,R90,R60,R45}

that will take a point and rotate it 45◦ around the point (2, 3). Sketchthis situation and explain your reasoning.


{Fx=0,Fy=0,Fy=x,R90,R60,R45}

that will take a point and rotate it 90◦ clockwise around the point (−3, 4)Sketch this situation and explain your reasoning.

20


1.3 Symmetry

Question What is symmetry?

You should think about the question above before reading on—though,uncharistically, we will give an answer eventually. Let’s start by giving someexamples of symmetry. An object can have symmetry through reflections:

Check it out—if the nose is along line line x = 0, then the reflection Fx=0 doesnot change the picture.

An object can have symmetry through rotations:

Check it out—if this object is centered at the origin, then the rotation R120 doesnot change the picture.

An object can have symmetry through translations if it extends infinitely insome direction, in this case imagine the pattern below extending infinitely bothto the left and the right:

Now the translation T(4,0) does not change the picture—note, we assumed thateach square above is 1 unit.

21

1.3. SYMMETRY

Question What other translation don’t change the picture?

?An object can have symmetry through dilations, here we need to imagine the

entire plane filled with tiles as below:

This picture (assuming it extended infinitely) would have symmetry through Dsfor some scale factor s, see Activity A.3.

Trying to combine all these different ideas of symmetry is not easy. In [7],it is said

Symmetry is immunity to a possible change.

Question Can you explain what the definition of symmetry above is saying?

?

1.3.1 Symmetry Groups

One of the most fundamental notions in all of modern mathematics is that of agroup. Sadly, many students never see a group in their education.

Definition A group is a set of elements (in our case matrices) which we willcall G such that:

(1) There is an associative operation (in our case matrix multiplication).

(2) The set is closed under this operation (the product of any two matrices inthe set is also in the set).

(3) There exists an identity I ∈ G such that for all M ∈ G,

IM = MI = M.

(4) For all M ∈ G there is an inverse M−1 ∈ G such that

MM−1 = M−1M = I.

22


1.3.2 Groups of Rotations

Let’s see a group. Here we have an equilateral triangle centered at the origin ofthe (x, y)-plane:

Question The matrix R360 will rotate this triangle completely around theorigin. What matrix will rotate this triangle one-third of a complete rotation?

As a gesture of friendship, I’ll take this one. One-third of 360 is 120. Sowe see that R120 will rotate the triangle one-third of a full rotation. Do youremember this matrix? Here it is:

R120 =

−12 −√3

2 0√32

−12 0

0 0 1

In spite of the fact that this matrix is messy and that matrix multiplication issomewhat tedious, you should realize that

R2120 = R240 and R3120 = R360.

Let’s put these facts (and a few more) together in what is called a group table.Remember multiplication tables from elementary school? Well, we’re going tomake something like a “multiplication table” of rotations. We’ll start by listingthe identity and powers of a one-third rotation along the top and left-hand sides.Setting R = R120 we have:

◦ I R R2 R3 · · ·

I

R

R2

R3

...

Since R3 = I, we need only take our table to R2. At this point we can write outthe complete table:

◦ I R R2

I I R R2

R R R2 I

R2 R2 I R

23

1.3. SYMMETRY

Since matrix multiplication is associative, and we see from the table that everyelement has an inverse, we see that

{I,R,R2}

is a group.

Question What rotation matrices would we use when working with a square?A pentagon? A hexagon?

?

1.3.3 Groups of Reflections

Let’s see another group. Again consider an equilateral triangle. This time weare interested in the three lines of reflection that preserve this triangle:

Question Suppose that the triangle above is centered at the origin of the(x, y)-plane. What are equations for ℓ, m, and n?

?The easiest of the reflections above is the reflection over Fℓ.

We’ll start our group table off with just two elements: I and F = Fℓ.

◦ I FI I FF F I

24


Notice that when we apply F twice we’re right back where we started. Hence,FF = I. Since matrix multiplication is associative, we see that

{I,F}

forms a group. Specifically this is a group of reflections of the triangle.

Question Above we used Fℓ. What would happen if we used Fm or Fn? Also,what are the equations for the lines of symmetry of the square centered at theorigin?

?

1.3.4 Symmetry Groups

Now let’s mix our rotations and reflections. Consider our original triangle andapply FℓR120:

What you may not immediately notice is that we obtain the same transformationby taking the original triangle and applying Fm.

As it turns out, every possible symmetry of the equilateral triangle canbe represented using reflections and rotations. Each of these reflections androtations can be expressed as a composition of a single reflection and a singlerotation. The collection of all symmetries forms a group called the symmetrygroup of the equilateral triangle.

Let’s see the group table, note we’ll let R = R120 and F = Fℓ:

25

1.3. SYMMETRY

◦ I R R2 F FR FR2

I I R R2 F FR FR2

R R R2 I FR2 F FR

R2 R2 I R FR FR2 F

F F FR FR2 I R R2

FR FR FR2 F R2 I R

FR2 FR2 F FR R R2 I

This table shows every symmetry of the triangle, including the identify I. Bycomparing the rows and columns of the group table, you can see that every ele-ment has an inverse. This combined with the fact that the matrix multiplicationis associative shows that the symmetries of the triangle,

{I,R,R2,F,FR,FR2}

form a group.

Question Can you express the symmetries of squares in terms of reflectionsand rotations? What does the group table look like for the symmetry group ofthe square?

?

26



(1) Explain what symmetry is.

(2) State the definition of a group of matrices.

(3) How many lines of reflectional symmetry exist for a square? Provide adrawing to justify your answer.

(4) What are the equations for the lines of reflectional symmetry that existfor the square? Explain your answers.

(5) How many lines of reflectional symmetry exist for a regular hexagon?Provide a drawing to justify your answer.

(6) What are the equations for the lines of reflectional symmetry for a regularhexagon? Explain your answers.

(7) How many consecutive rotations are needed to return the vertexes of asquare to their original position? Provide a drawing to justify your answer,labeling the vertexes.

(8) How many degrees are in one-fourth of a complete rotation of the square?Explain your answer.

(9) How many degrees are in one-sixth of a complete rotation of the regularhexagon? Explain your answer.

(10) In this section, we’ve focused on a 3-sided figure, a 4-sided figure, and a6-sided figure. Why do we not include the rotation group for the pentagonin this section? If we did, how many degrees would be in one-fifth of acomplete rotation?

(11) With notation used in this section, draw pictures representing the actionof the following isometries Fℓ, R, RFℓ and FℓR on the equilateral triangle.

(12) Consider a square centered at the origin. Draw pictures representing theaction of Fy=0, R90, R90Fy=0, and Fy=0R90 on this square.

(13) Consider a hexagon centered at the origin. Draw pictures representing theaction of Fx=0, R60, R60Fx=0, and Fx=0R60 on this hexagon.

(14) Find two symmetries of the equilateral triangle, neither of which is theidentity, such that their composition is R120. Explain and illustrate youranswer.

(15) Find two symmetries of the equilateral triangle, neither of which is theidentity, such that their composition is Fx=0. Explain and illustrate youranswer.

27

1.3. SYMMETRY

(16) Find two symmetries of the equilateral triangle, neither of which is theidentity, such that their composition is R2120. Explain and illustrate youranswer.

(17) Find two symmetries of the square, neither of which is the identity, suchthat their composition is R180. Explain and illustrate your answer.

(18) Find two symmetries of the square, neither of which is the identity, suchthat their composition is Fℓ. Explain and illustrate your answer.

(19) Find two symmetries of the square, neither of which is the identity, suchthat their composition is R270. Explain and illustrate your answer.

(20) Use a group table to help you write out the symmetries of the equilateraltriangle. List all elements that commute with every other element in thetable. Explain your reasoning.

(21) Use a group table to help you write out the symmetries of the square. Listall elements that commute with every other element in the table. Explainyour reasoning.

(22) Use a group table to help you write out the symmetries of the regularhexagon. List all elements that commute with every other element in thetable. Explain your reasoning.

(23) Let M be a symmetry of the equilateral triangle. Define

C(M) = {all symmetries that commute with M}.

Write out C(M) for every symmetry M of the equilateral triangle. Makesome observations.

28

Chapter 2

Folding and TracingConstructions

We don’t even know if Foldspace introduces us to one universe ormany. . .

—Frank Herbert

2.1 Constructions

While origami as an art form is quite ancient, folding and tracing constructionsin mathematics are relatively new. The earliest mathematical discussion offolding and tracing constructions that I know of appears in T. Sundara Row’sbook Geometric Exercises in Paper Folding, [8], first published near the endof the Nineteenth Century. In the Twentieth Century it was shown that everyconstruction that is possible with a compass and straightedge can be done withfolding and tracing. Moreover, there are constructions that are possible viafolding and tracing that are impossible with compass and straightedge alone.This may seem strange as you can draw a circle with a compass, yet this seemsimpossible to do via paper-folding. We will address this issue in due time. Let’sget down to business—here are the rules of folding and tracing constructions:

Rules for Folding and Tracing Constructions

(1) You may only use folds, a marker, and semi-transparent paper.

(2) Points can only be placed in two ways:

(a) As the intersection of two lines.

(b) By marking “through” folded paper onto a previously placed point.Think of this as when the ink from a permanent marker “bleeds”through the paper.

29

2.1. CONSTRUCTIONS

(3) Lines can only be obtained in three ways:

(a) By joining two points—either with a drawn line or a fold.

(b) As a crease created by a fold.

(c) By marking “through” folded paper onto a previously placed line.

(4) One can only fold the paper when:

(a) Matching up points with points.

(b) Matching up a line with a line.

(c) Matching up two points with two intersecting lines.

Now we are going to present several basic constructions. We will proceed bythe order of difficulty of the construction.

Construction (Transferring a Segment) Given a segment, we wish to moveit so that it starts on a given point, on a given line.

Construction (Copying an Angle) Given a point on a line and some angle,we wish to copy the given angle so that the new angle has the point as its vertexand the line as one of its edges.

Transferring segments and copying angles using folding and tracing withouta “bleeding marker” can be tedious. Here is an easy way to do it:

Use 2 sheets of paper and a pen that will mark through multiplesheets.

Question Can you find a way to do the above constructions without using amarker whose ink will pass through paper?

?Construction (Bisecting a Segment) Given a segment, we wish to cut it inhalf.

(1) Fold the paper so that the endpoints of the segment meet.

(2) The crease will bisect the given segment.

Question Which rule for folding and tracing constructions are we usingabove?

30

CHAPTER 2. FOLDING AND TRACING CONSTRUCTIONS

?Construction (Perpendicular through a Point) Given a point and a line, wewish to construct a line perpendicular to the original line that passes throughthe given point.

(1) Fold the given line onto itself so that the crease passes though the givenpoint.

(2) The crease will be the perpendicular line.


?Construction (Bisecting an Angle) We wish to divide an angle in half.

(1) Fold a point on one leg of the angle to the other leg so that the creasepasses though the vertex of the angle.

(2) The crease will bisect the angle.


?Construction (Parallel through a Point) Given a line and a point, we wish toconstruct another line parallel to the first that passes through the given point.

(1) Fold a perpendicular line through the given point.

(2) Fold a line perpendicular to this new line through the given point.

31

2.1. CONSTRUCTIONS

Now there may be a pressing question in your head:

Question How the heck are we going to fold a circle?

First of all, remember the definition of a circle:

Definition A circle is the set of points that are a fixed distance from a givenpoint.

Question Is the center of a circle part of the circle?

?Secondly, remember that when doing compass and straightedge construc-

tions we can only mark points that are intersections of lines and lines, linesand circles, and circles and circles. Thus while we technically draw circles, wecan only actually mark certain points on circles. When it comes to folding andtracing constructions, drawing a circle amounts to marking points a given dis-tance away from a given point—that is exactly what we can do with compassand straightedge constructions.

Construction (Intersection of a Line and a Circle) We wish to construct thepoints where a given line meets a given circle. Note: A circle is given by a pointon the circle and the central point.

(1) Fold the point on the circle onto the given line so that the crease passesthrough the center of the circle.

(2) Mark this point though both sheets of paper onto the line.


?

32


Question How could you check that your folding and tracing construction iscorrect?

?Construction (Equilateral Triangle) We wish to construct an equilateral tri-angle given the length of one side.

(1) Bisect the segment.

(2) Fold one end of the segment onto the bisector so that the crease passesthrough the other end of the segment. Mark this point onto the bisector.

(3) Connect the points.

Question Which rules for folding and tracing constructions are we usingabove?

?Construction (Intersection of Two Circles) We with to intersect two circles,each given by a center point and a point on the circle.

(1) Use four sheets of tracing paper. On the first sheet, mark the centers ofboth circles. On the next two sheets, mark the center and point on eachof the circle—one circle per sheet.

(2) Simply move the two sheets with the centers and points on the circles, sothat the centers are over the centers from the first sheet, and the pointson the circles coincide. Now on the fourth sheet, mark all points.

?Question How do you use folding and tracing to construct a regular hexagon?What other regular polygons can you construct?

?

33

2.1. CONSTRUCTIONS


(1) What are the rules for folding and tracing constructions?

(2) Use folding and tracing to bisect a given line segment. Explain the stepsin your construction.

(3) Given a line segment with a point on it, use folding and tracing to constructa line perpendicular to the segment that passes through the given point.Explain the steps in your construction.

(4) Use folding and tracing to bisect a given angle. Explain the steps in yourconstruction.

(5) Given a point and line, use folding and tracing to construct a line parallelto the given line that passes through the given point. Explain the stepsin your construction.

(6) Given a point and line, use folding and tracing to construct a line perpen-dicular to the given line that passes through the given point. Explain thesteps in your construction.

(7) Given a circle (a center and a point on the circle) and line, use folding andtracing to construct the intersection. Explain the steps in your construc-tion.

(8) Given a line segment, use folding and tracing to construct an equilateraltriangle whose edge has the length of the given segment. Explain the stepsin your construction.

(9) Explain how to use folding and tracing to transfer a segment.

(10) Given an angle and some point, use folding and tracing to copy the angleso that the new angle has as its vertex the given point. Explain the stepsin your construction.

(11) Explain how to use folding and tracing to construct envelope of tangentsfor a parabola.

(12) Use folding and tracing to construct a square. Explain the steps in yourconstruction.

(13) Use folding and tracing to construct a regular hexagon. Explain the stepsin your construction.

(14) Morley’s Theorem states: If you trisect the angles of any triangle withlines, then those lines form a new equilateral triangle inside the original

34


triangle.

Give a folding and tracing construction illustrating Morley’s Theorem.Explain the steps in your construction.

(15) Given a length of 1, construct a triangle whose perimeter is a multiple of6. Explain the steps in your construction.

(16) Construct a 30-60-90 right triangle. Explain the steps in your construc-tion.

(17) Given a length of 1, construct a triangle with a perimeter of 3 +√5.

Explain the steps in your construction.

35

2.2. ANATOMY OF FIGURES

2.2 Anatomy of Figures

Remember, in studying geometry we seek to discover the points that can beobtained given a set of rules. Now the set of rules consists of the rules forfolding and tracing constructions.

Question In regards to folding and tracing constructions, what is a point?

?Question In regards to folding and tracing constructions, what is a line?

?Question In regards to folding and tracing constructions, what is a circle?

?OK, those are our basic figures, pretty easy right? Now I’m going to quiz

you about them (I know we’ve already gone over this, but it is fundamental sojust smile and answer the questions):

Question Place two points randomly in the plane. Do you expect to be ableto draw a single line that connects them?

?Question Place three points randomly in the plane. Do you expect to beable to draw a single line that connects them?

?Question Place two lines randomly in the plane. How many points do youexpect them to share?

?Question Place three lines randomly in the plane. How many points do youexpect all three lines to share?

?Question Place three points randomly in the plane. Will you (almost!) al-ways be able to draw a circle containing these points? If no, why not? If yes,how do you know?

?

36


2.2.1 Lines Related to Triangles

Believe it or not, in mathematics we often try to study the simplest objects asdeeply as possible. After the objects listed above, triangles are among the mostbasic of geometric figures, yet there is much to know about them. There areseveral lines that are commonly associated to triangles. Here they are:

• Perpendicular bisectors of the sides.

• Bisectors of the angles.

• Altitudes of the triangle.

• Medians of the triangle.

The first two lines above are self-explanatory. The next two need definitions.

Definition An altitude of a triangle is a line segment originating at a vertexof the triangle that meets the line containing the opposite side at a right angle.

Definition A median of a triangle is a line segment that connects a vertexto the midpoint of the opposite side.

Question The intersection of any two lines containing the altitudes of a tri-angle is called an orthocenter. How many orthocenters does a given trianglehave?

?Question The intersection of any two medians of a triangle is called a cen-troid. How many centroids does a given triangle have?

?Question What is the physical meaning of a centroid?

?

2.2.2 Circles Related to Triangles

There are also two circles that are commonly associated to triangles. Here theyare:

• The circumcircle.

• The incircle.

These aren’t too bad. Check out the definitions.

37


Definition The circumcircle of a triangle is the circle that contains all threevertexes of the triangle. Its center is called the circumcenter of the triangle.

Question Does every triangle have a circumcircle?

?Definition The incircle of a triangle is the largest circle that will fit insidethe triangle. Its center is called the incenter of the triangle.

Question Does every triangle have an incircle?

?Question Are any of the lines described above related to these circles and/orcenters? Clearly articulate your thoughts.

?

38



(1) In regards to folding and tracing constructions, what is a circle? Compareand contrast this to a naive notion of a circle.

(2) Place three points in the plane. Give a detailed discussion explaining howthey may or may not be on a line.

(3) Place three lines in the plane. Give a detailed discussion explaining howthey may or may not intersect.

(4) Explain how a perpendicular bisector is different from an altitude. Drawan example to illustrate the difference.

(5) Explain how a median is different from an angle bisector. Draw an exampleto illustrate the difference.

(6) What is the name of the point that is the same distance from all threesides of a triangle? Explain your reasoning.

(7) What is the name of the point that is the same distance from all threevertexes of a triangle? Explain your reasoning.

(8) Could the circumcenter be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(9) Could the orthocenter be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(10) Could the incenter be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(11) Could the centroid be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(12) Are there shapes that do not contain their centroid? If so, draw a pictureand explain. If not, explain why not using pictures as necessary.

(13) Draw an equilateral triangle. Now draw the lines containing the altitudesof this triangle. How many orthocenters do you have as intersections oflines in your drawing? Hints:

(a) More than one.

(b) How many triangles are in the picture you drew?

(14) Given a triangle, construct the circumcenter. Explain the steps in yourconstruction.

(15) Given a triangle, construct the orthocenter. Explain the steps in yourconstruction.

39


(16) Given a triangle, construct the incenter. Explain the steps in your con-struction.

(17) Given a triangle, construct the centroid. Explain the steps in your con-struction.

(18) Given a triangle, construct the incircle. Explain the steps in your con-struction.

(19) Given a triangle, construct the circumcircle. Explain the steps in yourconstruction.

(20) Where is the circumcenter of a right triangle? Explain your reasoning.

(21) Where is the orthocenter of a right triangle? Explain your reasoning.

(22) Can you draw a triangle where the circumcenter, orthocenter, incenter,and centroid are all the same point? If so, draw a picture and explain. Ifnot, explain why not using pictures as necessary.

(23) True or False: Explain your conclusions.

(a) An altitude of a triangle is always perpendicular to a line containingsome side of the triangle.

(b) An altitude of a triangle always bisects some side of the triangle.

(c) The incenter is always inside the triangle.

(d) The circumcenter can be outside the triangle.

(e) The orthocenter is always inside the triangle.

(f) The centroid is always inside the incircle.

(24) Given 3 distinct points not all in a line, construct a circle that passesthrough all three points. Explain the steps in your construction.

(25) The following picture shows a triangle that has been folded along thedotted lines:

Explain how the picture “proves” the following statements:

(a) The interior angles of a triangle sum to 180◦.

(b) The area of a triangle is given by bh/2.

(26) Use folding and tracing to construct a triangle given the length of one side,the length of the the median to that side, and the length of the altitudeof the opposite angle. Explain the steps in your construction.

40


(27) Use folding and tracing to construct a triangle given one angle, the lengthof an adjacent side and the altitude to that side. Explain the steps in yourconstruction.

(28) Use folding and tracing to construct a triangle given one angle and thealtitudes to the other two angles. Explain the steps in your construction.

(29) Use folding and tracing to construct a triangle given two sides and thealtitude to the third side. Explain the steps in your construction.

41

2.3. SIMILAR TRIANGLES

2.3 Similar Triangles

In geometry, we may have several different segments all with the same length.We don’t want to say that the segments are equal because that would mean thatthey are exactly the same—position included. Hence we need a new concept:

Definition When different segments have the same length, we say they arecongruent segments.

Definition In a similar fashion, if there is an isometry that maps one angle tothe another angle, then we say they are congruent angles.

Put these together and we have the definition for triangles:

Definition Two triangles are said to be congruent triangles if there is anisometry that maps one triangle to the other triangle.

After working with triangles for short time, one quickly sees that the notionof congruence is not the only “equivalence” one wants to make between triangles.In particular, we want the notion of similarity.

One day when aloof old Professor Rufus was trying to explain similar trian-gles to his class, he merely wrote

△ABC ∼ △A′B′C ′ ⇔∠A ≃ ∠A′∠B ≃ ∠B′∠C ≃ ∠C ′

and walked out of the room.

Question Can you give 3 much needed examples of similar triangles?

?Question Devise a way to use folding and tracing constructions to help ex-plore this notion of similar triangles.

?Another day when aloof old Professor Rufus was trying to explain similar

triangles to his class, he merely wrote

△ABC ∼ △A′B′C ′ ⇔AB = k ·A′B′BC = k ·B′C ′CA = k · C ′A′

and walked out of the room.

Question Can you give 3 much needed examples of similar triangles?

?

42


Question Devise a way to use folding and tracing constructions to help ex-plore this notion of similar triangles.

?Question What’s going on in aloof old Professor Rufus’ head1—why are hisexplanations so different?

Well in fact, both definitions of similar triangles given above are equivalent.

Definition Two triangles△ABC and△A′B′C ′ are said to be similar if eitherequivalent condition holds:

∠A ≃ ∠A′∠B ≃ ∠B′∠C ≃ ∠C ′

orAB = k ·A′B′BC = k ·B′C ′CA = k · C ′A′

Let’s see if we can put similar triangles to work. Here is an old carpenter’strick. Suppose you have two parallel lines.

If you want to make two more lines that divide the space into three equal regions,you can place a ruler diagonally across the parallel lines, and then mark of one-third of the diagonal distance.

From this you can get your three regions.

1I realize that this is a dangerous question!

43


Question Can you explain why this works using similar triangles?

?Now for a more challenging situation. Suppose we have an equilateral tri-

angle of side length 2. Use similar triangles and the Pythagorean Theorem tofind the length of the segment a (the segment that goes from the center of thetriangle to a side at a right angle) in the picture below.

a

2

Question How do you do this?

?

44



(1) Compare and contrast the ideas of equal triangles, congruent triangles,and similar triangles.

(2) Explain why all equilateral triangles are similar to each other.

(3) Explain why all isosceles right triangles are similar to each other.

(4) Explain why when given a right triangle, the altitude of the right angledivides the triangle into two smaller triangles each similar to the originalright triangle.

(5) The following sets contain lengths of sides of similar triangles. Solve forall unknowns—give all solutions. In each case explain your reasoning.

(a) {3, 4, 5}, {6, 8, x}(b) {3, 3, 5}, {9, 9, x}(c) {5, 5, x}, {10, 4, y}(d) {5, 5, x}, {10, 8, y}(e) {3, 4, x}, {4, 5, y}

(6) A Pythagorean Triple is a set of three positive integers {a, b, c} such thata2 + b2 = c2. Write down an infinite list of Pythagorean Triples. Explainyour reasoning and justify all claims.

(7) Here is a right triangle, note it is not drawn to scale:

Solve for all unknowns in the following cases.

(a) a = 3, b =?, c =?, d = 12, e = 5, f =?

(b) a =?, b = 3, c =?, d = 8, e = 13, f =?

(c) a = 7, b = 4, c =?, d =?, e = 11, f =?

(d) a = 5, b = 2, c =?, d = 6, e =?, f =?

In each case explain your reasoning.

(8) Suppose you have two similar triangles. What can you say about the areaof one in terms of the area of the other? Be specific and explain yourreasoning.

45


(9) During a solar eclipse we see that the apparent diameter of the Sun andMoon are nearly equal. If the Moon is around 24000 miles from Earth,the Moon’s diameter is about 2000 miles, and the Sun’s diameter is about865000 miles how far is the Sun from the Earth?

(a) Draw a relevant (and helpful) picture showing the important pointsof this problem.

(b) Solve this problem, be sure to explain your reasoning.

(10) When jets fly above 8000 meters in the air they form a vapor trail. Cruisingaltitude for a commercial airliner is around 10000 meters. One day Ireached my arm into the sky and measured the length of the vapor trailwith my hand—my hand could just span the entire trail. If my hand spans9 inches and my arm extends 25 inches from my eye, how long is the vaportrail? Explain your reasoning.



(11) David proudly owns a 42 inch (measured diagonally) flat screen TV.Michael proudly owns a 13 inch (measured diagonally) flat screen TV.Dave sits comfortably with his dog Fritz at a distance of 10 feet. How farmust Michael stand from his TV to have the “same” viewing experience?Explain your reasoning.



(12) You love IMAX movies. While the typical IMAX screen is 72 feet by 53feet, your TV is only a 32 inch screen—it has a 32 inch diagonal. Howclose do you have to sit to your screen to simulate the IMAX format?Explain your reasoning.



(13) David proudly owns a 42 inch (measured diagonally) flat screen TV.Michael proudly owns a 13 inch (measured diagonally) flat screen TV.Michael stands and watches his TV at a distance of 2 feet. Dave sits com-fortably with his dog Fritz at a distance of 10 feet. Whose TV appearsbigger to the respective viewer? Explain your reasoning.


46



(14) Here is a personal problem: Suppose you are out somewhere and you seethat when you stretch out your arm, the width of your thumb is the sameapparent size as a distant object. How far away is the object if you knowthe object is:

(a) 6’ long (as tall as a person).

(b) 16’ long (as long as a car).

(c) 40’ long (as long as a school bus).

(d) 220’ long (as long as a large passenger airplane).

(e) 340’ long (as long as an aircraft carrier).


(15) I was walking down Woody Hayes Drive, standing in front of St. JohnArena when a car pulled up and the driver asked, “Where is Ohio Sta-dium?” At this point I was a bit perplexed, but nevertheless I answered,“Do you see the enormous concrete building on the other side of the streetthat looks like the Roman Colosseum? That’s it.”

The person in the car then asked, “Where are the Twin-Towers then?”Looking up, I realized that the towers were in fact just covered by top ofOhio Stadium. I told the driver to just drive around the stadium untilthey found two enormous identical towers—that would be them. Theythanked me and I suppose they met their destiny.

I am about 2 meters tall, I was standing about 100 meters from the OhioStadium and Ohio Stadium is about 40 meters tall. If the Towers arearound 500 meters from the rotunda (the front entrance of the stadium),how tall could they be and still be obscured by the stadium? Explain yourreasoning—for the record, the towers are about 80 meters tall.

(16) Consider the following combinations of S’s and A’s. Which of them pro-duce a Congruence Theorem? Which of them produce a Similarity Theo-rem? Explain your reasoning.

SSS, SSA, SAS, SAA, ASA, AAA

47


(17) Explain how the following picture “proves” the Pythagorean Theorem.

ab

a

bc

b2

ac ac

b

c

a2

c2

bc

48

Chapter 3

Another Dimension

Another dimension, another dimension, another dimension, anotherdimension, another dimension, another dimension, another dimen-sion, another dimension, another dimension, another dimension, an-other dimension, another dimension!

—The Beastie Boys

3.1 Tessellations

Go to the internet and look up M.C. Escher. He was an artist. Look at some ofhis work. When you do your search be sure to include the word “tessellation”OK? Back already? Very good. Sometimes Escher worked with tessellations.What’s a tessellation? I’m glad you asked:

Definition A tessellation is a pattern of polygons fitted together to coverthe entire plane without overlapping.

While it is impossible to actually cover the entire plane with shapes, if wegive you enough of a tessellation, you should be able to continue it’s patternindefinitely. Here are pieces of tessellations:

On the left we have a tessellation of a square and an octagon. On the right wehave a “brick-like” tessellation.

Definition A tessellation is called a regular tessellation if it is composed ofcopies of a single regular polygon and these polygons meet vertex to vertex.

49

3.1. TESSELLATIONS

Example Here are some examples of regular tessellations:

Johannes Kepler, who lived from 1571–1630, was one of the first people tostudy tessellations. He certainly knew the next theorem:

Theorem 3 There are only 3 regular tessellations.

Question Why is the theorem above true?

?Since one can prove that there are only three regular tessellations, and we

have shown three above, then that is all of them. On the other hand there arelots of nonregular tessellations. Here are two different ways to tessellate theplane with a triangle:

Here is a way that you can tessellate the plane with any old quadrilateral:

3.1.1 Tessellations and Art

How does one make art with tessellations? To start, a little decoration goes along way. Check this out: Decorate two squares as such:

50

CHAPTER 3. ANOTHER DIMENSION

Tessellate them randomly in the plane to get this lightning-like picture:

Question What sort of picture do you get if you tessellate these decoratedsquares randomly in a plane?

?Another way to go is to start with your favorite tessellation:

Then you modify it a bunch to get something different:

51

3.1. TESSELLATIONS

Question What kind of art can you make with tessellations?

?I’m not a very good artist, but I am a mathematician. So let’s use a tessel-

lation to give a proof! Let me ask you something:

Question What is the most famous theorem in mathematics?

Probably the Pythagorean Theorem comes to mind. Let’s recall the state-ment of the Pythagorean Theorem:

Theorem 4 (Pythagorean Theorem) Given a right triangle, the sum of thesquares of the lengths of the two legs equals the square of the length of thehypotenuse. Symbolically, if a and b represent the lengths of the legs and c isthe length of the hypotenuse,

ac

b

then

a2 + b2 = c2.

Let’s give a proof! Check out this tessellation involving 2 squares:

Question How does the picture above “prove” the Pythagorean Theorem?

Solution The white triangle is our right triangle. The area of the middleoverlaid square is c2, the area of the small dark squares is a2, and the area of

52


the medium lighter square is b2. Now label all the “parts” of the large overlaidsquare:

2

3

4

1

5

From the picture we see that

a2 = {3 and 4}b2 = {1, 2, and 5}c2 = {1, 2, 3, 4, and 5}

Hencec2 = a2 + b2

Since we can always put two squares together in this pattern, this proof willwork for any right triangle. ■

Question Can you use the above tessellation to give a dissection proof of thePythagorean Theorem?

?

53

3.1. TESSELLATIONS


(1) Show two different ways of tessellating the plane with a given scalenetriangle. Label your picture as necessary.

(2) Show how to tessellate the plane with a given quadrilateral. Label yourpicture.

(3) Show how to tessellate the plane with a nonregular hexagon. Label yourpicture.

(4) Give an example of a polygon with 9 sides that tessellates the plane.

(5) Give examples of polygons that tessellate and polygons that do not tes-sellate.

(6) Give an example of a triangle that tessellates the plane where both 4 and8 angles fit around each vertex.

(7) True or False: Explain your conclusions.

(a) There are exactly 5 regular tessellations.

(b) Any quadrilateral tessellates the plane.

(c) Any triangle will tessellate the plane.

(d) If a triangle is used to tessellate the plane, then it is always the casethat exactly 6 angles will fit around each vertex.

(e) If a polygon has more than 6 sides, then it cannot tessellate the plane.

(8) Given a regular tessellation, what is the sum of the angles around a givenvertex?

(9) Given that the regular octagon has 135 degree angles, explain why youcannot give a regular tessellation of the plane with a regular octagon.

(10) Fill in the following table:

Regular Does it Measure If it tessellates, hown-gon tessellate? of an angle many surround each vertex?

3-gon4-gon5-gon6-gon7-gon8-gon9-gon10-gon

Hint: A regular n-gon has interior angles of 180(n− 2)/n degrees.

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(a) What do the shapes that tessellate have in common?

(b) Make a graph with the number of sides of an n-gon on the horizontalaxis and the measure of a single angle on the vertical axis. Brieflydescribe the relationship between the number of sides of a regularn-gon and the measure of one of its angles.

(c) What regular polygons could a bee use for building hives? Give somereasons that bees seem to use hexagons.

(11) Considering that the regular n-gon has interior angles of 180(n − 2)/ndegrees, and Problem (10) above, prove that there are only 3 regulartessellations of the plane.

(12) Explain how the following picture “proves” the Pythagorean Theorem.

55

3.2. SOLIDS

3.2 Solids

3.2.1 Platonic Solids

Around 2400 years ago, there was a group of mystics—today we might call it acult—who called themselves the Pythagoreans. Being mystics, the Pythagore-ans had some strange ideas, but on the other hand they were an enlightenedgroup of people, because they believed that they could better understand theuniverse around them by studying mathematics. As part of the Pythagoreans’numerological religion, they thought that some polyhedra had special powers.The Pythagoreans associated the following polyhedra to elements of nature asfollows:

Fire: Tetrahedron Air: Octahedron

Earth: Cube Water: Icosahedron

While the idea that these polyhedra are somehow connected to “arcane elementsof nature” is complete nonsense, there is actually something special about thesolids above. They are regular convex polyhedra. Let’s dissect those last words:

Definition A polyhedron is a three dimensional solid that is bounded by afinite number of polygons.

Definition An object is convex if given any two points inside the object, thesegment connecting those two points is also contained inside the object.

Definition A polyhedron is regular if all its faces are the same regular poly-gon and if the same number of polygons meet at every vertex.

Apparently the Pythagoreans discovered the above four regular convex poly-hedra first. Only later did they discover a fifth, the dodecahedron:

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Æther: Dodecahedron

Since the first four regular convex solids had elements associated to them, thePythagoreans reasoned that this fifth solid must also be associated to an el-ement. However, all the elements were accounted for. So the Pythagoreansassociated the dodecahedron to what we might call the æther, a mysteriousnon-earthly substance.

Question Consider the triangular dipyramid, the solid where two tetra-hedrons are joined at a face:

Is this a regular convex polyhedron? Explain your reasoning.

?Question Are there other regular convex polygons other than the ones shownabove?

We’ll give you the answer to this one:

Theorem 5 There are at most 5 regular convex polyhedra.

Proof To start, a corner of a three-dimensional object made of polygons musthave at least 3 faces. Now start with the simplest regular polygon, an equilateraltriangle. You can make a corner by placing:

(1) 3 triangles together.

57

3.2. SOLIDS



Thus each of the above configuration of triangles could give rise to a regular con-vex polyhedra. However, you cannot make a corner out of 6 or more equilateraltriangles, as 6 triangles all connected lie flat on the plane.

Now we can make a corner with 3 square faces, but we cannot make a cornerwith 4 or more square faces as 4 or more squares lie flat on the plane.

Finally we can make a corner with 3 faces each shaped like a regular pen-tagon, but we cannot make a corner with 4 or more regular pentagonal faces asthey will be forced to overlap.

Could we make a corner with 3 faces each shaped like a regular hexagon? No,because any number of hexagons will lie flat on the plane. A similar argumentworks to show that we cannot make a corner with 3 faces shaped like a regularn-gon where n > 6, except now instead of the shapes lying flat on the plane,they overlap.

Thus we could at most have 5 regular convex solids. ■

Question Above we prove that there are at most 5 regular convex polyhedra.How do we know that these actually exist?

?The five regular convex polyhedra came to be known as thePlatonic Solids,

over 2000 years ago when Plato discussed them in his work Timaeus. Here is aempty table of facts about the Platonic Solids—could you be so kind and fill itin?

Solid Vertexes Edges Faces

tetrahedron

octahedron

cube

icosahedron

dodecahedron

These solids have haunted men for centuries. People who were trying tounderstand the universe wondered what was the reason that there were only5 regular convex polyhedra. Some even thought there was something specialabout certain numbers.

Question In what ways does the number 5 come up in life that might lead aperson into believing it is a special number? Does this mean that the number5 is more special than any other number?

?

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(1) Explain what a polyhedron is.

(2) Explain what it means for an object to be convex.

(3) Explain what it means for a polyhedron to be regular.

(4) State which of the following are convex sets:

(a) A hollow sphere.

(b) A half-space.

(c) The intersection of two spherical solids.

(d) A solid cube.

(e) A solid cone.

(f) The union of two spherical solids.


(5) Use words and pictures to describe the following objects:

(a) A tetrahedron.

(b) An octahedron.

(c) A cube.

(d) An icosahedron.

(e) A dodecahedron.

(f) A triangular dipyramid.

(6) How is a regular convex polyhedron different from any old convex polyhe-dron?

(7) True or False:

(a) There are only 5 convex polyhedra.

(b) The icosahedron has exactly 12 faces.

(c) Every vertex of the tetrahedron touches exactly 3 faces.

(d) The octahedron has exactly 6 vertexes.

(e) Every vertex of the dodecahedron touches exactly 3 faces.


(8) While there are only 5 regular convex polyhedra, there are also nonconvexregular polyhedra. Draw some examples.

(9) Draw picture illustrating the steps of the proof of Theorem 5.

(10) Where does the proof of Theorem 5 use the fact that the solids are convex?

59

3.2. SOLIDS

(11) A dual polyhedron is the polyhedron obtained when one connects thecenters of all the pairs of adjacent faces of a given polyhedron. What arethe dual polyhedra of the Platonic Solids? Explain your reasoning.

(12) How many rotational symmetries does the tetrahedron have? Explainyour reasoning.

(13) How many rotational symmetries does the cube have? Explain your rea-soning.

(14) How many rotational symmetries does the octahedron have? Explain yourreasoning.

(15) How many rotational symmetries does the dodecahedron have? Explainyour reasoning.

(16) How many rotational symmetries does the icosahedron have? Explainyour reasoning.

(17) Consider a cube whose side has a length of 1 unit. Imagine an octahedroninside this cube whose vertexes are the centers of the faces of the cube.How long are the edges of this new octahedron?

(18) Consider a tetrahedron whose side has a length of 1 unit. Imagine anothertetrahedron inside whose vertexes are the centers of the faces of the originaltetrahedron. How long are the edges of this new tetrahedron?

(19) Consider a cube whose side has a length of 1 unit. Imagine an octahedronaround this cube so that the centers of the faces of the octahedron are onthe vertexes of the cube. How long are the edges of this new octahedron?

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3.3 Higher Dimensions

Consider this picture:

Ceci n’est pas un cube.

Here, the French got it right. That is not a cube—at best it is what a shadow ofa real cube might look like on this sheet of paper. Being that paper is effectively2-dimensional, it is seemingly impossible to have an actual cube embedded inthis text. The miracle is that when we look at this shadow of a cube, we knowthat it represents a real cube. You may not think this is much of a miracle, let’ssee if I can convince you otherwise. Meet Sugarbear:

Just as the image at the beginning of this section was a picture of a three-dimensional cube, this is not actually Sugarbear—rather this is a picture ofSugarbear who is also actually 3-dimensional.

Sugarbear has decided to take a vacation to Flatland. The curious readershould see [1] for a history of Flatland. In Flatland there are only two direc-tions left/right and up/down. Flatlanders (as they’re called) have no concept ofinward/outward. As it so happens, you already know someone from Flatland,Louie Llama! Here’s Louie Llama chillin in front of his house:

61

3.3. HIGHER DIMENSIONS

Notice, that even when Louie Llama is standing comfortably inside his housewith the door closed, you can still see inside his house! This is a privilege wereceive for living in the third-dimension.

On the other hand, Louie Llama cannot see you. From his perspective, he issurrounded by four walls, nothing could possibly get inside or out of his house.So our favorite Llama was quite confused when Sugarbear greeted him with ajolly “Hello!” as she arrived in Flatland.

Remember, Louie Llama can only see stuff in the plane that is Flatland. FromLouie Llama’s perspective, Sugarbear’s “Hello!” came from all around him andeven from inside himself! So to help out the puzzled llama, Sugarbear decidedto take the plunge into Flatland, actually jumping through the plane that LouieLlama lives in. When she does this, Louie Llama sees something quite strange.Our llama sees a series of cross-sections of Sugarbear:

First Second Third

Fourth Fifth Sixth

However, each cross-section is very confusing and seems to morph into thenext. This wasn’t very helpful for our friend Louie Llama.

Question Can you make sense of what Louie Llama saw?

62


?Sugarbear now says “Greetings from the third-dimension.” Louie Llama is

quite perplexed—he only knows two dimensions. To help him out Sugarbeartries to tell him what a cube is.

Question Can you help Sugarbear describe what a cube is to a two-dimensionalFlatlander? How might Louie Llama react to a real cube?

?At this point, Sugarbear gets an idea. Why not “pop” Louie Llama out of

the plane so that he can see Flatland the way we do? Being rather rambunctiousfor a bear, Sugarbear decides to do this sending Louie Llama off into the airfloating above Flatland. This is a very strange experience for Louie Llama.

As he floats above Flatland, he can see his world in a way that he had never seenbefore. He can see inside closed building and even inside his friends! Finally, hedrifts back down into Flatland. From his friends’ perspective he just “appeared”out of nowhere. When his friends ask him where he came from, he can onlyreply, “up” but cannot show them where this is. Louie Llama’s worldview haschanged forever.

At the end of her vacation in Flatland, Sugarbear turns to you and asks youthe following question:

If Flatlanders had no idea that there could be a third dimension,who’s to say that we are not as blind as they are. Could there be afourth dimension?

Question What would you say to Sugarbear? Be careful not to upset her,her claws are quite sharp.

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?While I honestly don’t know if there is a real fourth dimension, we can work

by analogy to figure out what it would be like should it actually exist. We’llstart with dimension 0 and see how high we can work up to.

Zero Dimensions A 0-dimensional object is just a point. Draw a point inthe box below:

One Dimension A 1-dimensional object is a segment, what you get fromconnecting two points. Draw two points connected by a segment below:

Two Dimensions There are many different 2-dimensional objects, we’ll makea square. To do this, simply copy the segment you drew above and connectcorresponding end-points. Let’s see it in the box below:

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Three Dimensions If you’re playing along at home, and I sincerely hope youare, you may realize that we are going to have a bit of trouble at this point. Nomatter, instead of drawing a real cube, we’ll draw a “shadow” of a cube. To dothis, simply copy the square you drew before, place it next to the original butoffset a tad. Then connect corresponding vertexes. Let’s see your shadow of thecube in the box below:

Four Dimensions Wow, if we had trouble with three-dimensions, what dowe do now? The answer is simple. We do the same thing we’ve done before.Simply copy your shadow of the cube and connect corresponding vertexes. Let’ssee it in the box below:

Pretty strange eh? This is called a hypercube.

Higher Dimensions We need not stop at dimension four, but as a gestureof friendship—we will. If we had continued on, we would make a shadow of ahyperhypercube! That would be pretty mind-blowing.

Some Observations and Thoughts Living in our three-dimensional worldas we do, it truly seems utterly impossible that there could be other dimen-sions. Yet, it is starting to seem that perhaps our common sense isn’t so sen-sible. Equipped with the tool of mathematics, physicist are discovering thatour universe is filled with symmetry. The symmetries we are finding cannotbe explained as the symmetry of a 2-dimensional or 3-dimensional objects—toexplain the observed phenomenon, we need higher dimensions.

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(1) For each of the activities below, explain if they are essentially 1-dimensional,2-dimensional, or 3-dimensional.

(a) Driving a car down the freeway.

(b) Riding on a roller coaster.

(c) Driving a car in a parking-lot.

(d) Driving a snow-machine.

(e) Flying an airplane.

(f) Swimming underwater.


(2) Suppose you stuck your fingers into Flatland. Describe what a Flatlanderwould see.

(3) Suppose a 4-dimensional creature stuck its “fingers” into our three dimen-sional space. What do you think we would see?

(4) Do you know what bubble tea is? If not, look it up. It is drunk with alarge straw that can be used to suck up tapioca balls. Some people lovebubble tea, others hate it. If you were a tapioca ball in a bubble tea straw,you could only move forward and backward, like this:

How many dimensions would you view your world in? How many dimen-sions would your world actually have? Explain your reasoning.

(5) Fill in the following table:

Vertexes Edges Faces Solids

PointSegmentSquareCube

HypercubeHyperhypercube

(6) Someone once told me that a simple way to draw a hypercube was to drawa regular octagon and then simply make squares on the inside of each edgeof the octagon. Does this work? Explain your reasoning.

(7) We’ve seen how to draw n-dimensional cubes. Explain how to draw n-dimensional triangles. Draw a triangle, a tetrahedron, and a hypertetra-hedron. Explain your reasoning.

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(8) An n-sphere is the set of points in n-dimensions that are all equidistantfrom a given point. Using this definition, explain why:

(a) A 2-sphere is a circle.

(b) A 3-sphere is a sphere.

In addition, explain what a 1-sphere would be and do your best to describea 4-sphere.

(9) Consider a square whose side has a length of 2 units.

(a) Compute its perimeter and area.

(b) Increase the side-length of this square by a factor of 3. What is thenew perimeter and area?


(10) Consider a cube whose side has a length of 2 units.

(a) Compute its perimeter, area, and volume.

(b) Increase the side-length of this square by a factor of 3. What is thenew perimeter, area, and volume?


(11) Consider a hypercube whose side has a length of 2 units.

(a) Compute its perimeter, area, volume, and hypervolume.

(b) Increase the side-length of this square by a factor of 3. What is thenew perimeter, area, volume, and hypervolume?


(12) If one can only travel along edges, what is the maximum distance betweentwo vertexes of:

(a) A square who side has length 1 unit.

(b) A cube who side has length 1 unit.

(c) A hypercube who side has length 1 unit.

(d) A hyperhypercube who side has length 1 unit.

(e) An n-dimensional cube who side has length 1 unit.


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Chapter 4

String Art

The artist is a receptacle for the emotions that come from all overthe place: from the sky, from the earth, from a scrap of paper, froma passing shape, from a spider’s web.

—Pablo Picasso

4.1 I’m Seeing Stars

Can you remember when you first learned to draw a star? I can! I was firsttaught to draw a star like this:

But I wanted to know how to draw 5-pointed stars like the other kids. So oneday I taught myself to draw a star like this:

This may seem like a silly way to draw this star—it certainly makes me chucklenow. Let’s see if we can give a theory for drawing stars that will connect thetwo different stars above, teach you how to draw some new stars, and learn alittle more mathemati

re ections on mathematics · 2012. 12. 13. · chapter 1 isometries and since you know you cannot...

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